DEPARTMENT OF AERONAUTICAL AND AUTOMOBILE ENGINEERING

MANIPAL INSTITUTE OF TECHNOLOGY

MANIPAL

V SEMESTER

AIRCRAFT SYSTEMS

LABORATORY RECORD

NAME :

BATCH :

ROLL NUMBER :

REGISTER NUMBER :

VI SEMESTER AERONAUTICAL

AERODYNAMICS AND PROPULSION LAB

Index page

SINO

NAME OF THE EXPERIMENT DATE MANUAL MARKS

OBSERVATION BOOK MARKS

SIGNATURE OF THE FACULTY

1 DEFLECTION OF A SIMPLY

SUPPORTED BEAM

2 VERIFICATION OF MAXWELL’S

RECIPROCAL THEOREM

3 DETERMINATION OF YOUNG’S

MODULUS USING STRAIN

GAUGE

4 POISSON’S RATIO

DETERMINATION

5 VERIFICATION OF

SUPERPOSITION THEOREM

6 BUCKLING LOAD OF SLENDER

ECCENTRIC COLUMNS AND

CONSTRUCTION OF

SOUTHWELL PLOT

7 SHEAR FAILURE OF BOLTED

AND RIVETED JOINTS

8 WAGNER BEAM

9

10

11

TOTAL MARKS

TOTAL ATTENDENCE

1. DEFLECTION OF A SIMPLY SUPPORTED BEAM

AIM:

To determine the deflection of a simply supported beam

EQUIPMENT:

Beam test set-up with load cells

Steel scale caliper

Flat beam

THEORY:

A beam shown in FIG-1 shows the section which is simply supported at the ends and is subjected to bending about its major axis with a concentrated load anywhere in the beam. The beam is provided with strain gauge, the deflection of the beam can be determined whenever the load is applied on the beam. Strain gauge values may be noted for several further works.

Deflections are given by following expressions; students may derive the expressions using unit load method or castigliano’s theorem.

yx = wb(L2-b2)x-x3/ 6EIL for 0<x<a

yx= Wb[x3-L/b(x-a)3-(L2-b2)x]/6EIL for a<x<L

Where W is the load applied at a distance ‘a’ from the left support in Newton.

Ixx= bd3/12= (25*63)/12= 450mm4

yx = wb(L2-b2)x-x3/ 6EIL = 9.81*250[(6502-2502)200-2003]/6*2*105*450*650 = 0.44

L = Span of the beam in mm Yx = deflection at any point x from left end Ixx = Moment of Inertia of the beam in mm4

E = Young’s Modulus in N/mm2

PROCEDURE:

Find the moment of inertia of beam from the following expression.

(1/12)bd3, where ‘b’ is width of beam and ‘d’ is depth

Place the beam supporting from wedge supports. The load position can be varied. Set the load cell to read zero in the absence of load. Set the deflection gauge to read zero in the absence of load. Load the beam with kg. Note the deflections before and after the load point through deflection gauge. Increase the load up to max 5kg in case of MS steel and up to max 3 kg in case of Aluminium and repeat the experiment.

RESULT:

The deflections of a simply supported beam at x mm from various loads are as follows:

2. VERIFICATION OF MAXWELL’S RECIPROCAL THEOREM

AIM: To Verify the maxwell’s theorem for the structures system.

EQUIPMENT: Beam Test Setup with Load Cells Steel Rule Caliper Flat Beam

THEORY:

The displacement at point i, in a linear elastic structure, due to concentrated load at point j is equal to the displacement at point j due to concentrated load of same magnitude at point i.

The displacement at each point will be measured in the direction of the concentrated load at that point. The only other restrictions on this statement, in addition to the structure being linear elastic and stable, is that the displacement at either point is/must be consistent with the type of load at that point. If the load at a point is a concentrated force, then the displacement at that point will be a translation, while if the load is moment, then the displacement will be rotation. The displacement at any point will be in the same direction as the load.

This theorem often referred to as Maxwell’s Reciprocal Theorem.

This can be proved through unit load method i.e, the deflection at A due to unit load at B is equal to deflection at B due to unit load at A.

Where, M = Bending Moment at any point x due to external load m = Bending moment at any point x due to unit load applied at the point where

deflection is required.

Let mXA = Bending Moment at any point x due to unit load at A mXB = Bending Moment at any point x due to unit load at B when unit load

(external load) is applied at A,

M = mXA

When unit load (external load) is applied at A, apply unit load at B. Then m=mXA

To find deflection at B due to unit load at A, apply unit load at B. Then m=mXB

Hence,

Similarly, when unit load (external load) is applied at B, M=mXB

To find deflection at A, then m=mXA

Hence,

Comparing equation 1 and 2,

The external load (w) can be taken as a multiple with unit load, therefore , the load w will appear as multiple with mXA in equation 1 and as multiple with mXB in equation 2

Thereby resulting in

A beam shown in figure below which is simply supported at the ends and is subjected to bending about its major axis with a concentrated load anywhere in the beam.

Deflections δx at any distance ‘X’ from left support are given by following expressions; reference may be made to experiment No.1

Where W is the load palced at a distance ‘a’ from the left support in newton.

b= distance of load from right side support L= span of the beam in mm Yx= deflection at any point distance x from the left end I= moment of inertia of the beam in mm4 (Ixx) E= young’s modulus in N/mm2

The Maxwell’s reciprocal displacement theorem is very useful in the analysis of statistically indeterminate structures for evaluating the flexibility co-efficient. The displacement relationship can be expressed at point i and j δi,B = fi,j w ......................................................................................(5) δj,A= fj,i w ......................................................................................(6)

where fij is the displacement at point i due to a unit load at point j and f j,i is the displacement at point j due to a unit load at point i. If we now substitute these expressions in Betti’s law and cancel out the term w on each side, we obtain

fij = fji

The theorem can be restated as the displacement at point i, an elastic structure, due to a unit load at point j is equal to the displacement at point j due to unit load at point i.

PROCEDURE:

Find the moment of inertia of beam from the following expression.

MOI = 1/12 b1d13

Where b1 is width of beam and d1 is depth. Place the beam supporting from two wedge supports. The load position can be varied. Set the load cell to read zero in the absence of load. Set the deflection gauge to read zero in the absence of load. Load in beam with 2.5 Kg. Note deflections at any point through deflection gauge. Interchange the load location with the point of deflection measurement and repeat the readings, increase the load to 5 Kg and repeat the experiment. Find deflections from the formula and verify.

RESULT: Hence Maxwell’s reciprocal theorem is verified.

Since, δAB = δBA

3 DETERMINATION OF YOUNG’S MODULUS USING STRAIN GAUGE

AIM:To determine the young’s modulus of a simply supported beam or a cantilever beam.

THEORY:A beam shown in fig-1 shows the section which is simply supported at the ends and is subjected to bending about its major axis with a concentrated load anywhere in the beam. The beam is provided with strain gauge, the deflection of the beam can be determined wherever the load is applied on to the beam. An available cantilever beam can be also utilized for this experiment.

The strain gauge is at a fixed position in the beam and load position can be varied. A strain gauge is mounted on a free surface, which in general is in a state of plane stress where the state of stress is with regards to a specific xy rectangular rosette. Consider the three element rectangular rosette shown in fig which provides normal strain components in three directions spaced at angles of 45°. If an xy coordinates system is assumed to coincide with the gauge A and C then εx = εA and εy = εc. Gauge B provides information necessary to determine γxy. Once εx,εy and γxy are known, then Hooke’s law can be used to determine σx,σy,ζxy. However in this case the requirement is to determine young’s modulus (E), which can be determined from equation (1) below

M/I = σ/yAlsoShear Modulus G = ζ xy/ γxy ........................................................(2)

Young’s Modulus E = 2G/ 1+γ

PROCEDURE: Mount the cantilever beam at the left support of beam test setup. Connect the strain gauges wire with the strain measuring equipment. Use the following colour codes.

εA = red and white wire -- Linear guageεB = black and blue wire -- Lateral guageεC = green and yellow wire -- 45o

Set the load cell to read Zero value in the absence of load. Set the three strains to read zero in the absence of load. Now load the beam with 2.5kg at some time and record the strains in three directions. Record the load value at the load cell.

Repeat the experiment with load value of 5kg. Compute the values of Poisson’s ratio from

γ= εy/εx

RESULT:

The young’s modulus is

4. Poisson’s ratio Determination

AIM: To determine the Poisson’s ratio of cantilever beam.

EQUIPMENT:Beam test set-up with load cells, cantilever beam with calibrated rosette strain gauge, strain measuring equipment.

THEORY:A cantilever beam is subjected to bending about its major axis with a concentrated load anywhere in the beam. The beam is subjected with rosette strain gauge.A calibrated strain gauge rosette is fixed at a location within the span of the beam, and load position can be varied. Calibration has been done to read strain in microns (μ). Consider the three elements rectangular rosette shown in fig. which provides normal strain components in three directions spaced at angle of 45 deg. If an xy coordinate system is assured to coincide with the gauge A & C then εx = εA & εy = εc

Gauge B provides information necessary to determine the shear strain (γxy). Once εx, εy, & γxy are shown, then Hooke’s law can be used to determine σx, σy & ζxy. Subsequently principal stresses can be determined.

PROCEDURE:Mount the cantilever beam at the left support of beam test setup. Connect the strain gauges wires with the strain measuring equipment. Use the following colour codes

εA = red and white wire -- Linear guageεB = black and blue wire -- Lateral guageεC = green and yellow wire -- 45o

set the load cell to read zero value in the absence of load. Set the three strains to read zero in the absence of load. Now load the beam with 2.5kg at some point and record the strains in three directions. Record the experiment with load value of 5kg.Compute the values of Poisson’s ratio from

γ = εy/εx

RESULT:The Poisson ratio is

5. VERIFICATION OF SUPERPOSITION THEOREM

AIM: To verify the theorem of super position.

EQUIPMENT: Beam test setup with multiple loading capability ( at least 2 loading points are required) 2 load cells, cantilever strain gauged beam. Strain measurement equipment.

THEORY: Many times, a structural member is subjected to a number of forces acting not only at the ends, but also at the intermediate points along its length. Such a member can be analyzed by the application of the principle of the superposition. The resulting strain will be equal to the algebraic sum of the strain caused by individual forces acting along the length of the member.

The strain gauge is at a fixed position in the beam and load position can be varied. A strain gauge is mounted on a free stream surface, which in general is in a state of plane stress where the state of stress with regard to a specific XY rectangular rosette. Consider the three element rectangular rosette shown in the figure, which provides normal strain component in three direction spaced at angles of 45 degree. If an XY co-ordinates system is assumed to coincide with the gauge A and C then єX=єA and єY=єC

Gauge B provides information necessary to determine ζXY.

PROCEDURE:

Mount the cantilever beam at the left support of beam test setup. Connect the strain gauges wires with the strain measuring equipment. Use the following color codes.

εA = red and white wire -- Linear guageεB = black and blue wire -- Lateral guageεC = green and yellow wire -- 45o

Set the load cell to read zero value in the absence id load. Set the three strain gauge to read zero in the absence of load. Now load the beam with 3.0 kg at some point from vertical and record the strain in three direction. Record the load value at the load cell. Record the point of loading.

Remove the load. Set the load cell to read zero value in the absence of the load. Set the three strains to read zero in the absence of the load. Now load the beam with load from horizontal direction. Apply, load from right side end with load value of 2.0kg and record the strains in three directions. Record the load value.

Remove this load. Set the load cell to read zero value in the absence of the load. Set the three strains to read zero in the absence of the load. Now load the beam with3.0kg at same point from vertical as done earlier.

In addition load the beam with load from horizontal direction. Apply load from right side end with load value of 2.0kg

Record the strain in three directions. Record the load values in the two load cells.

record the vertical load position.

Compute the values of ζXY from the formula.

ζXY = 2 єB-єA -єC

RESULT: The values of δAB=δBA. Hence superposition theorem is verified.

6. BUCKLING LOAD OF SLENDER ECCENTRIC COLUMNS AND CONSTRUCTION OF SOUTH WELL PLOT

AIM: Practical columns have some imperfections in the form of initial curvature and the buckling of loads of such struts is of real practical values. This experiment aims at measuring the buckling load and construction of south well plot. The imperfection amounts to initial curvature, which shows up in this plot.

EQUIPEMENT: WAGNER Beam setup, hinged supports, load cells, long column, with initial curvature, mounted dial for deflection measurements.

THEORY: Consider a pin strut AB of length L, whose centroidal longitudinal axis is initially curved under the application of the end load P, the strut will have some additional lateral displacement y at any section.

The value of Pe represents the buckling load for perfectly straight strut. In the relation for deflection (y), the additional lateral displacement of the strut, that the effect of end load P is to increase(y) by a factor 1/(Pe/P)-1; shown by eqn(6) when P approaches Pe, the addition displacement at mid length of the strut is expressed by eqn(7).

The load deflection relationship of eqn(5) is the basic of south well plot technique for extrapolating for the elastic critical load from experimental measurement.

The linear relationship between δp & δ shown in figure below can be experimentally determined. Thus if a straight line is drawn which best fits the points determined from the experimental measurement of P and δ, the reciprocal of the slope of this line gives an estimate that can be determined from the intercept on the horizontal axis.

PROCEDURE:

Setup the 2 hinge supports on the Wagner beam at the top and bottom supports. Fix the column in the support.

Set the load reading to zero in the load cell Determine the centre of column. Setup the deflection dialguage for reading

the column deflection at the centre column Set the deflection dial guage reading to zero. Apply the vertical load in steps of 5 kgs. Each in four steps (5,10,15,20kgs)

and record the deflection at each step of load.

Result:

7. SHEAR FAILURE OF BOLTED AND RIVITED JOINTS

AIM: To determine the ultimate shear stress on the bolt.

EQUIPMENT: Wagner beam setup, bolt for failure analysis, vernier caliper.

THEORY: Riveted & bolted connections are common in structural assemblies. Following are the modes of failure in riveted joint.

Tension Failure Shearing failure across one or more planes of rivet Bearing failure between plates and the rivet. Plate shear or shear out failure in the plate

In a riveted joint, the rivets may themselves fail in shear. The tendency is to cut through the rivet across the section lying is said to be in shear lying in the plane between plates it connected.

If the load is transmitted through the bearing between the plates and the shank of the rivet producing shear in the rivet is said to be in shear.

When the load is transmitted by shear in only one section of the rivet, the rivet is said to be in single shear. When the loading of the rivet is such as to have the load transmitted in two shear planes, the rivet is said to be in double shear. When the load is transmitted in more than 2 planes the rivet is said the be in multiple shear.

Rivets and bolts subjected to both shear and axial tension shall be so proportional that the calculated shear and axial tension do not exceed the allowable stress ζvf & σtf

and the following expression does not exceed the specific value.

SHEARING FAILURE OF THE RIVET: In riveted joint, the rivets may themselves fail in shear. The tendency is to cut through the rivet across the section lying in the plane between the plates it connect.

In analysis this possible manner of failure one must always note whether a rivet acts in single shear or double shear. The latter case, the two cross-sectional areas of the same rivet resist the applied force. The shearing stress is assumed to be uniformaly distributed over the cross section of the rivet.

Let Pus = Pull required, per pitch length, for shear failure.

Fs = Ultimate shear strength of the rivet material.

d = gross diameter of the bolt.

PROCUDURE:

1. Setup the WAGNER beam test setup.2. Note the diameter of the bolt.3. Place the bolt in the slot.4. Set the load cell to read zero.5. Apply the load gradually.6. Note the load reading at the point of shear failure of mild steel bolt.

RESULT:

Shear stress for single shear = N/mm2

Shear stress for double shear = N/mm2

8. WAGNER BEAM (TENSION FIELD).

AIM: The objective of this experiment is to determine the constant K of the wagner beam.

INTRODUCTION:

In the analysis of wing beams of airplanes, the designer is faced with several problems which, in general are not present in civil engineering structural design. The civil engineer endeavors to make the web sheet of all beams thick enough so that the web not buckle before the design load is reached on the structure.

Buckling is a case of failure and the shearing stress causing buckling determines the allowable maximum shear stress is given by equation (1).

If the sheet is very thin, buckling stress is given by equation (1) is extremely low and, in the interest of making efficient use of all available material. The aircraft engineer raises the question as to how much additional shear can be carried by such a buckled plate before.

Some portion of the sheet has a total stress equal to the yield point of the material, thus giving rise to permanent deformations: or

The ultimate strength is reached.

If now, we assume that the web plate in the beam is very thin the above discussion of principle stresses. We see that one of them is compressive stress against which thin plates have a very low resistance. The tendency will then be for the plate to buckles in a direction perpendicular to the compressive stress at a value of the applied shear which becomes less as the web becomes thinner and thinner, the limiting case being for a sheet of zero thickness.

In this case, the sheet buckles upon the application of a shear load and can only resist shear by means of the tensile stresses will tend to pull the two beam flanges together, thus necessitating vertical members to counteract this tendency.

Wagner assumes that the web buckles immediately upon the application of shear load and that the only stresses resisting the shear forces are the tensile stresses which act approximately 45 degree.

Considering infinitely rigid parallel span flanges and vertical stiffners the following equation for this limiting case is shown in equation 2,3,4and 5.

The limiting case of a web having no compressive strength has been treated by wagner and beams approximating this are known as wagner beam. The basic assumption of this theory is that total shear force in the web can be divided into a shear force in the sheet and the shear forced carried by diagonal tension;

K= (1- ζcr / ζ)n

A strain gauge is mounted in a free surface, which in general, is in a state of plane stress where the state if stress with regards to a specific xy rectangular rosette since each gauge element provides only one piece of information, the indicated normal strain at the point in direction of the gauge. Consider the three elements rectangular rosette shown in the figure which provides normal strain components in three directions spaced at angles of 45 degrees.

In an XY coordinate system is assumes to coincide with the gauge A and C then

єX=єA and єY=єC are known, then Hooke’s law can be used to determine σX, σY, ζXY